The new dinv is not so new
Abstract
In (Duane, Garsia, Zabrocki 2013) the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in (Hicks, Kim 2013) a nonrecursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in (Haglund, Remmel, Wilson 2018). Moreover, we give also a noncompositional proof of the "$ehh$" case of the shuffle conjecture (after (Garsia, Xin, Zabrocki 2014)) by bijectively proving a relation with the two part case of the Delta conjecture.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.02912
 Bibcode:
 2019arXiv190302912D
 Keywords:

 Mathematics  Combinatorics;
 05E05
 EPrint:
 17 pages, 8 figures